Calculus PYQ — Page 22

The value of the integral ${\int }_{1}^{2}(\frac{{t}^{4}+1}{{t}^{6}+1})dt$ is :

Let $f(x)=x+\frac{a}{{\pi }^{2}-4}\mathrm{sin}x+\frac{b}{{\pi }^{2}-4}\mathrm{cos}x$, $x\in \mathbb{R}$ be a function which satisfies $f(x)=x+{\int }_{0}^{\pi /2}\mathrm{sin}(x+y)f(y)dy$. Then $(a+b)$ is equal to

The integral $16{\int }_{1}^{2}\frac{dx}{{x}^{3}{({x}^{2}+2)}^{2}}$ is equal to

Let $T$ and $C$ respectively, be the transverse and conjugate axes of the hyperbola $16{x}^{2}-{y}^{2}+64x+4y+44=0$. Then the area of the region above the parabola ${x}^{2}=y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is:

Area of the region ${(x,y):{x}^{2}+{(y-2)}^{2}\leq 4,{x}^{2}\geq 2y}$ is

Let $\alpha$ be the area of the larger region bounded by the curve ${y}^{2}=8x$ and the lines $y=x$ and $x=2$, which lies in the first quadrant. Then the value of $3\alpha$ is equal to

Let $x=x(y)$ be the solution of the differential equation $2(y+2){\mathrm{log}}_{e}(y+2)dx+(x+4-2{\mathrm{log}}_{e}(y+2))dy=0$, $y>-1$ with $x({e}^{4}-2)=1$. Then $x({e}^{9}-2)$ is equal to

The set of all $a\in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is

Let $f(x)=\frac{\mathrm{sin}x+\mathrm{cos}-\sqrt{2}}{\mathrm{sin}x-\mathrm{cos}x},x\in [0,\pi ]-{\frac{\pi }{4}}$, then $f(\frac{7\pi }{12}){f}^{"}(\frac{7\pi }{12})$ is equal to

$\underset{n\rightarrow \infty }{\mathrm{lim}}{({2}^{\frac{1}{2}}-{2}^{\frac{1}{3}})({2}^{\frac{1}{2}}-{2}^{\frac{1}{5}})....({2}^{\frac{1}{2}}-{2}^{\frac{1}{2n+1}})}$ is equal to

Let $f(x)=\int \frac{dx}{(3+4{x}^{2})\sqrt{4-3{x}^{2}}},|x|<\frac{2}{\sqrt{3}}$. If $f(0)=0$ and $f(1)=\frac{1}{\alpha \beta }{\mathrm{tan}}^{-1}(\frac{\alpha }{\beta }),\alpha ,\beta >0$, then ${\alpha }^{2}+{\beta }^{2}$ is equal to _______.

Let $f(x)$ be a function satisfying $f(x)+f(\pi -x)={\pi }^{2},\forall x\in \mathbb{R}$. Then ${\int }_{0}^{\pi }f(x)\mathrm{sin}xdx$ is equal to

If $A$ is the area in the first quadrant enclosed by the curve $C:2{x}^{2}-y+1=0$, the tangent to $C$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60A$ is................

Let the function $f:[0,2]\rightarrow \mathbb{R}$ be defined as $f(x)={\begin{matrix}{e}^{\mathrm{min}{{x}^{2},x-[x]}}, & x\in [0,1) \\ {e}^{[x-{\mathrm{log}}_{e}x]}, & x\in [1,2]\end{matrix}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral ${\int }_{0}^{2}xf(x)dx$ is

If the area of the region bounded by the curves ${y}^{2}-2y=-x$ and $x+y=0$ is $A$, then $8A=$

Let $I(x)=\int \sqrt{\frac{x+7}{x}}dx$ and $I(9)=12+7{\mathrm{log}}_{e}7$. If $I(1)=\alpha +7{\mathrm{log}}_{e}(1+2\sqrt{2}),$ then ${\alpha }^{4}$ is equal to _____.

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation ${x}^{2}-px+\frac{5}{4}p=0$ are rational. Then the area of the region ${(x,y):0\leq y\leq (x-q{)}^{2},0\leq x\leq q}$ is

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{48}{{x}^{4}}{\int }_{0}^{x}\frac{{t}^{3}}{{t}^{6}+1}dt$ is equal to

The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is

Let $I(x)=\int \frac{x+1}{x{(1+x{e}^{x})}^{2}}dx,x>0.$ If $\underset{x\rightarrow \infty }{\mathrm{lim}}I(x)=0$ then $I(1)$ is equal to

If $y(x)={x}^{x},x>0$, then ${y}^{"}(2)-2{y}^{'}(2)$ is equal to :

The area of the region enclosed by $y\leq 4{x}^{2},{x}^{2}\leq 9y$ and $y\leq 4$, is equal to

The general solution of the differential equation $(x-{y}^{2})dx+y(5x+{y}^{2})dy=0$ is